Say that we specify a model of the Bollen political democracy data and draw posterior samples using the following blavaan code (where save.lvs saves the latent variable samples for further use):

  model <- '
# latent variable definitions
ind60 =~ x1 + x2 + x3
dem60 =~ y1 + a*y2 + b*y3 + c*y4
dem65 =~ y5 + a*y6 + b*y7 + c*y8

# regressions
dem60 ~ ind60
dem65 ~ ind60 + dem60

# residual correlations
y1 ~~ y5
y2 ~~ y4 + y6
y3 ~~ y7
y4 ~~ y8
y6 ~~ y8
'

fit <- bsem(model, data=PoliticalDemocracy, save.lvs = TRUE)

We describe here how to summarize the fitted model. The most obvious functions are summary(), coef(), and vcov(), which all work in a manner similar to the analogous lavaan functions. But instead of maximum likelihood estimates and standard errors, blavaan reports posterior means and posterior standard deviations. Other summaries that are unique to Bayesian models include model convergence metrics, model fit/comparison metrics, and samples of latent variables. These are discussed below.

### Convergence

Following model estimation, we immediately wish to look at the “goodness” of the posterior samples, including convergence to a stationary distribution and autocorrelation. Popular convergence metrics are available via the blavInspect() function:

blavInspect(fit, 'rhat')
blavInspect(fit, 'neff')

where R-hat values near 1.00 indicate convergence, and large effective sample sizes (hundreds or above) are preferred. For details on these metrics, see, e.g., the Posterior Analysis section of the Stan Reference Manual.

### Model Fit & Comparison

Next, we may wish to examine some model fit metrics. While many metrics are available from the summary() output, more are available from the fitMeasures() function:

summary(fit)
fitMeasures(fit)

For judging absolute fit, blavaan supplies a posterior predictive p-value that is based on the likelihood ratio statistic. Good-fitting models have values near 0.5 on this metric. For examining models’ relative fits, blavaan supplies the DIC, WAIC, and LOOIC. The latter two metrics are computed with the help of the loo package (Vehtari et al. 2020). Comparison of multiple models on these criteria is facilitated via blavCompare(), which provides standard errors of the difference between two criteria.

Other notable functions include blavFitIndices() for alternative measures of absolute fit and ppmc() for general posterior predictive checks.

### Latent Variables & Standardization

An often-discussed advantage of Bayesian models is their abilities to describe uncertainty in “random” parameters, including random effects and latent variables. To access this functionality in blavaan, users must set save.lvs = TRUE during model estimation, as is done at the top of this page. After model estimation, uses can access this information via blavInspect() or blavPredict(). Relevant arguments to blavInspect() include lvmeans and lvs. The former returns posterior means of latent variables, which are similar to the predictions supplied by frequentist models. The latter returns posterior samples of latent variables, so that users could summarize their uncertainties or other functions of latent variables. These posterior samples are returned as a list of length n.chains, where each list entry has a row per posterior sample (and number of columns is total number of latent variables in the model):

postmns <- blavInspect(fit, what = "lvmeans")
postsamps <- blavInspect(fit, what = "lvs")

Some related, but different, information can be obtained by blavPredict(). This function will also return posterior samples of latent variables, but in a matrix instead of a list:

postsamps <- blavPredict(fit, type = "lv")

The blavPredict() function will also return predictions of observed variables conditioned on the sampled latent variables. The type = "yhat" argument returns expected values of observed variables conditioned on latent variable samples; the type = "ypred" argument returns posterior predictions of observed variables including residual noise (essentially yhat + error); and the type = "ymis" argument returns posterior predictions of missing variables conditioned on observed. These expected values and predictions are returned in list format; for a matrix, see the last line of code below.

evpreds <- blavPredict(fit, type = "yhat")
postpreds <- blavPredict(fit, type = "ypred")
mispreds <- blavPredict(fit, type = "ymis")

## convert to matrix from list:
evpreds <- do.call("rbind", evpreds)

Finally, not fully related to latent variables: the standardizedPosterior() function will return standardized posterior draws. It calls the lavaan function standardizedSolution() in the background and has some of that function’s flexibility.

### References

Vehtari, Aki, Jonah Gabry, Mans Magnusson, Yuling Yao, Paul-Christian Bürkner, Topi Paananen, and Andrew Gelman. 2020. “Loo: Efficient Leave-One-Out Cross-Validation and WAIC for Bayesian Models.” https://mc-stan.org/loo/.